Brute weak necessities
The two-dimensionalist account of a posteriori (metaphysical) necessity can be motivated by two observations.
First, all good examples of a posteriori necessities follow a priori from non-modal truths. For example, as Kripke pointed out, that his table could not have been made of ice follows a priori from the contingent, non-modal truth that the table is made of wood. Simply taking metaphysical modality as a primitive kind of modality would make a mystery of this fact.
Second, it is a well-known linguistic fact that there are two ways of evaluating a sentence at a possible context or scenario. In one sense, 'it is warmer than it is here' is false at every context; in another, the sentence is true at a context c iff it is warmer at c than at the actual utterance context @.
Developing the second observation leads to two-dimensional semantics, and further to the two-dimenational account of metaphysical modality. On that account, a sentence is a priori just in case it is true at all worlds when evaluated in the first way ("as actual"), and necessary just in case it is true at all worlds when evaluated in the second way ("as counterfactual"). The space of worlds is the same in either case.
The two kinds of evaluation come apart if and only if the sentence in question contains actually-dependent expressions such as indexicals ('here'), demonstratives ('this table'), or proper names. Evaluated as counterfactual, such expressions denote whatever plays a certain role in the actual world (the utterance context). This is why the empirical information one needs to know a posteriori necessities is ordinary non-modal information about the world.
The two-dimensionalist account assumes that all a posteriori necessities can be explained away by the difference between evaluating as actual and as counterfactual. Let's call those a posteriori necessities tame, and any others brute. For example, if 'there is an omniscient being' is an a posteriori necessity, it would be brute, since there is plausibly no difference between evaluating the sentences at worlds considered as actual and at worlds considered as counterfactual.
There are many reasons to be sceptical about brute necessites, besides the fact that there are no clear examples. Among other things, they would complicate our metaphysics; they would raise epistemic worries; and (my favourite reason) they would make the domain of metaphysical modality philosophically uninteresting. For example, it would be mysterious why anyone should care whether the mental supervenes on the physical within some brute sphere of "metaphysical" possibility.
Brute necessities are almost the same thing as what Chalmers calls strong necessities. According to Chalmers, a strong necessity is an a posteriori necessity that is true at all metaphysically possible worlds when evaluated as actual. Every strong necessity is brute, but not every brute necessity is strong.
To illustrate, consider the hypothesis that (G) is a strong necessity:
(G) Some omniscient being likes H2O.
That is, suppose that for reasons we can't explain from the armchair and through explorations into ordinary non-modal truths, every metaphysically possible world contains an omniscient being who likes H2O, and let's suppose 'H2O' picks out the same substance in worlds considered as actual and as counterfactual.
Next, assume that 'water' rigidly picks out whatever plays the water role in the actual world, and that H2O plays this role, so that water is necessarily H2O. If (G) is necessary and 'water=H2O' is necessary, then plausibly so is (G').
(G') Some omniscient being likes water.
In contrast to (G), (G') is not true at all metaphysically possible worlds considered as actual. For assume XYZ plays the water role at w, and the only omniscient being at w likes H2O but not XYZ. Then (G') is false at w considered as actual. (G') is not a strong necessity. But it is brute, for its necessity cannot be explained away along two-dimensional lines.
Now, (G') entails (G), so if there are no strong necessities, then there are also no brute necessities like (G'). So maybe it's enough to focus on strong necessities?
Arguably not. In the example, we get the strong-necessity (G) from (G') by replacing the actuality-dependent term 'water' with the non-actuality-dependent term 'H2O' which at any world w considered as actual picks out what 'water' denotes at w considered as counterfactual. But arguably we can't always find such a term.
Assume 'charge' rigidly denotes whatever microphysical quantity plays the physical role of charge, and that different quantities play that role at different worlds. (Not an uncontroversial assumption, but one that should be compatible with two-dimensionalism.) Then consider the hypothesis that (H') is a brute metaphysical necessity.
(H') Some omniscient being likes charge.
For the same reason for which (G') is not a strong necessity, (H') is not a strong necessity either. But this time, we have no "semantically neutral" way to pick out the quantity that plays the charge role, so we cannot convert (H') into a strong necessity (H).
(H) Some omniscient being likes xxx.
Moreover, it's not a coincidence that we don't have the required word. Try to introduce a word that would do the job! The word would somehow have to pick out the quantity that plays the charge role "by its inner nature" rather than by any role it plays. But we have no conception of what that inner nature is, and it's not even clear what it would mean to have such a conception.
Admittedly, the assumption that (H') is necessary still entails a strong necessity, namely that some omniscient being exists. But that hole is easily plugged:
(H'') Either there is no omniscient being or some omniscient being likes charge.
If the two-dimensional account of a posteriori necessity is correct, (H'') cannot be necessary. But (H'') is not a strong necessity, nor does the necessity of (H'') entail any strong necessity. So banning strong necessities is not enough.